Calculation of Local Bifurcation Points in Piecewise Nonlinear Discrete-Time Dynamical Systems

This paper proposes a method for calculation of local bifurcation points in discrete-time dynamical systems with piecewise nonlinear characteristics (PNDDS). First, an -dimensional PNDDS, which has two piecewise nonlinear maps, is shown and its variational equation is derived. Next, a calculation method for the local bifurcation points that utilizes the conditional equation for the periodic solution and the characteristic equation is proposed. It is essential to calculate the derivatives of the map with an initial value and with a bifurcation parameter to obtain the bifurcation points continuously in the parameter space. The above calculation process is a key component of the proposed method, and is explained in detail. Finally, we apply the proposed method to a two-dimensional PNDDS and calculate the local bifurcation points in order to confirm the validity of the proposed method.